The main aim of the current article is considering a nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law and deriving its approximate analytical solution in a systematic way. More precisely, after reformulating the nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law, its approximate analytical solution is derived formally through the use of the homotopy analysis transform method (HATM) which is based on the homotopy method and the Laplace transform. The existence and uniqueness of the solution of the nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law are also studied by adopting the fixed‐point theorem. In the end, by considering some two‐ and three‐dimensional graphs, the influence of the order of time‐fractional operator on the displacement is examined in detail.

中文翻译：

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带有米塔格-莱夫勒定律的非线性时分柯西反应扩散方程的近似解的解析研究

本文的主要目的是考虑具有Mittag-Leffler定律的非线性时分柯西反应扩散方程，并系统地推导其近似解析解。更精确地讲，在用米塔格-莱夫勒定律重新构造非线性时间分数阶柯西反应扩散方程之后，通过使用基于同伦方法和同伦方法的同伦分析变换方法（HATM）正式推导了其近似解析解。拉普拉斯变换。通过定点定理，研究了具有米特塔格-莱夫勒定律的非线性时分柯西反应扩散方程解的存在性和唯一性。最后，通过考虑一些二维和三维图，